Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

Structural basis of cytoskeletal regulation by twinfilin

The actin cytoskeleton is required, in all eukaryotic organisms, for several key cellular functions such as cell motility, cytokinesis, and endocytosis. In cells, actin exists either in a monomeric state (G-actin) or in a filamentous form (F-actin). F-actin is the functional form, which can assemble into various structures and produce direct pushing forces that are required for different motile processes. The assembly of actin monomers into complicated three-dimensional structures is tightly regulated by a large number of actin regulating proteins. One central actin regulating protein is twinfilin. Twinfilin consists of two actin depolymerizing-factor homology (ADF-H) domains, which are capable of binding actin, and is conserved from yeast to mammals. Previously it has been shown that twinfilin binds to and sequesters G-actin, and interacts with the heterodimeric capping protein. More recently it has been found that twinfilin also binds to the fast growing actin filament ends and prevents their growth. However, the cellular role of twinfilin and the molecular mechanisms of these interactions have remained unclear. In this study we characterized the molecular mechanisms behind the functions of twinfilin. We demonstrated that twinfilin forms a high-affinity complex with ADP-bound actin monomers (ADP-G-actin). Both ADF-H domains are capable of binding G-actin, but the C-terminal domain contains the high-affinity binding site. Our biochemical analyses identified twinfilin s C-terminal tail region as the interaction site for capping protein. Contrary to G-actin binding, both ADF-H domains of twinfilin are required for the actin filament barbed end capping activity. The C-terminal domain is structurally homologous to ADF/cofilin and binds to filament sides in a similar manner, providing the main affinity for F-actin during barbed end capping. The structure of the N-terminal domain is more distant from ADF/cofilin, and thus it can only associate with G-actin or the terminal actin monomer at the filament barbed end, where it regulates twinfilin s affinity for barbed ends. These data suggest that the mechanism of barbed end capping is similar for twinfilin and gelsolin family proteins. Taken together, these studies revealed how twinfilin interacts with G-actin, filament barbed ends, and capping protein, and also provide a model for how these activities evolved through a duplication of an ancient ADF/cofilin-like domain.

Studies of Immune Regulation in Type 1 Diabetes

Type 1 diabetes (T1D) is considered to be an autoimmune disease. In T1D insulin producing pancreatic β cells are destroyed. The disease process begins years before the clinical diagnosis of T1D. During the pathogenesis of T1D, pancreatic islets are infiltrated by cells of the immune system and T-lymphocytes are considered to be the main mediators of the β-cell destruction. In children with an active β-cell destruction process, autoantibodies against β-cell antigens appear in the blood. Individuals at increased risk of developing T1D can often be identified by detecting serum autoantibodies against β-cell antigens. Immunological aberrancies associated with T1D are related to defects in the polarization of T cells and in the function of regulatory mechanisms. T1D has been considered as an organ-specific autoimmune disease mediated by uncontrolled Th1-responses. In human T1D, the evidence for the role of over-expression of cytokines promoting cytotoxicity is controversial. For the past 15 years, regulatory T cells (Tregs) have been recognized as having a key role in the initiation and maintenance of tolerance, limiting harmful autoantigen-specific inflammation processes. It is possible that, if regulatory mechanisms fail to be initiated, the subtle inflammation targeting β cells lead to insulitis and eventually to overt T1D in some individuals. In the present thesis, we studied the induction of Tregs during the generation of T-cell responses in T1D. The results suggest that the generation of regulatory mechanisms and effector mechanisms upon T-cell activation is aberrant in children with T1D. In our studies, an in vitro cytotoxic environment inhibited the induction of genes associated with regulatory functions upon T-cell activation. We also found T1D patients to have an impaired cytotoxic response against coxsackievirus B4. Ineffective virus clearance may increase the apoptosis of β cells, and thus the risk of β-cell specific autoimmunity, due to the increased presentation of β-cell-derived peptides by APCs to T cells in pancreatic lymph nodes. Recently, a novel T helper cell subset called Th17 has been discovered. Animal models have associated Th17 cells and especially co-producers of IL-17 and IFN-γ with the pathogenesis of T1D. We aimed to characterize the role of Th17 immunity in human T1D. We demonstrated IL-17 activation to be a major alteration in T1D patients in comparison to healthy children. Moreover, alterations related to the FOXP3-mediated regulatory mechanisms were associated with the IL-17 up-regulation seen in T1D patients. These findings may have therapeutic implications for the treatment and prevention of T1D.